The challenging deployment of compute- and memory-intensive methods from Deep Neural Network (DNN)-based Continual Learning (CL) underscores the critical need for a paradigm shift towards more efficient approaches. Neuromorphic Continual Learning (NCL) appears as an emerging solution, by leveraging the principles of Spiking Neural Networks (SNNs) which enable efficient CL algorithms executed in dynamically-changed environments with resource-constrained computing systems. Motivated by the need for a holistic study of NCL, in this survey, we first provide a detailed background on CL, encompassing the desiderata, settings, metrics, scenario taxonomy, Online Continual Learning (OCL) paradigm, recent DNN-based methods to address catastrophic forgetting (CF). Then, we analyze these methods considering CL desiderata, computational and memory costs, as well as network complexity, hence emphasizing the need for energy-efficient CL. Afterward, we provide background of low-power neuromorphic systems including encoding techniques, neuronal dynamics, network architectures, learning rules, hardware processors, software and hardware frameworks, datasets, benchmarks, and evaluation metrics. Then, this survey comprehensively reviews and analyzes state-of-the-art in NCL. The key ideas, implementation frameworks, and performance assessments are also provided. This survey covers several hybrid approaches that combine supervised and unsupervised learning paradigms. It also covers optimization techniques including SNN operations reduction, weight quantization, and knowledge distillation. Then, this survey discusses the progress of real-world NCL applications. Finally, this paper provides a future perspective on the open research challenges for NCL, since the purpose of this study is to be useful for the wider neuromorphic AI research community and to inspire future research in bio-plausible OCL.